* [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
* [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
* [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
- * [VSW] Numerical evalutation of multiple polylogarithms, J.Vollinga, S.Weinzierl
+ * [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259
*
* - The order of parameters and arguments of Li and zeta is defined according to the nested sums
* representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
*/
/*
- * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <sstream>
// performs the actual series summation for multiple polylogarithms
cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
{
+ // ensure all x <> 0.
+ for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
+ }
+
const int j = s.size();
+ bool flag_accidental_zero = false;
std::vector<cln::cl_N> t(j);
cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
int q = 0;
do {
t0buf = t[0];
- // do it once ...
q++;
t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
for (int k=j-2; k>=0; k--) {
+ flag_accidental_zero = cln::zerop(t[k+1]);
t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
}
- // ... and do it again (to avoid premature drop out due to special arguments)
q++;
t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
for (int k=j-2; k>=0; k--) {
+ flag_accidental_zero = cln::zerop(t[k+1]);
t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
}
- } while (t[0] != t0buf);
+ } while ( (t[0] != t0buf) || flag_accidental_zero );
return t[0];
}
for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
if (!(*it).is_zero()) {
++depth;
- if (abs(*it) - y < -pow(10,-Digits+2)) {
+ if (abs(*it) - y < -pow(10,-Digits+1)) {
need_trafo = true;
- break;
}
if (abs((abs(*it) - y)/y) < 0.01) {
need_hoelder = true;
if (x.op(x.nops()-1).is_zero()) {
need_trafo = true;
}
- if (depth == 1 && !need_trafo) {
+ if (depth == 1 && x.nops() == 2 && !need_trafo) {
return -Li(x.nops(), y / x.op(x.nops()-1)).evalf();
}
+ // do acceleration transformation (hoelder convolution [BBB])
+ if (need_hoelder) {
+
+ ex result;
+ const int size = x.nops();
+ lst newx;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ newx.append(*it / y);
+ }
+
+ for (int r=0; r<=size; ++r) {
+ ex buffer = pow(-1, r);
+ ex p = 2;
+ bool adjustp;
+ do {
+ adjustp = false;
+ for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
+ if (*it == 1/p) {
+ p += (3-p)/2;
+ adjustp = true;
+ continue;
+ }
+ }
+ } while (adjustp);
+ ex q = p / (p-1);
+ lst qlstx;
+ lst qlsts;
+ for (int j=r; j>=1; --j) {
+ qlstx.append(1-newx.op(j-1));
+ if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
+ qlsts.append( s.op(j-1));
+ } else {
+ qlsts.append( -s.op(j-1));
+ }
+ }
+ if (qlstx.nops() > 0) {
+ buffer *= G_numeric(qlstx, qlsts, 1/q);
+ }
+ lst plstx;
+ lst plsts;
+ for (int j=r+1; j<=size; ++j) {
+ plstx.append(newx.op(j-1));
+ plsts.append(s.op(j-1));
+ }
+ if (plstx.nops() > 0) {
+ buffer *= G_numeric(plstx, plsts, 1/p);
+ }
+ result += buffer;
+ }
+ return result;
+ }
+
// convergence transformation
if (need_trafo) {
return result;
}
- // do acceleration transformation (hoelder convolution [BBB])
- if (need_hoelder) {
-
- ex result;
- const int size = x.nops();
- lst newx;
- for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
- newx.append(*it / y);
- }
-
- for (int r=0; r<=size; ++r) {
- ex buffer = pow(-1, r);
- ex p = 2;
- bool adjustp;
- do {
- adjustp = false;
- for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
- if (*it == 1/p) {
- p += (3-p)/2;
- adjustp = true;
- continue;
- }
- }
- } while (adjustp);
- ex q = p / (p-1);
- lst qlstx;
- lst qlsts;
- for (int j=r; j>=1; --j) {
- qlstx.append(1-newx.op(j-1));
- if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
- qlsts.append( s.op(j-1));
- } else {
- qlsts.append( -s.op(j-1));
- }
- }
- if (qlstx.nops() > 0) {
- buffer *= G_numeric(qlstx, qlsts, 1/q);
- }
- lst plstx;
- lst plsts;
- for (int j=r+1; j<=size; ++j) {
- plstx.append(newx.op(j-1));
- plsts.append(s.op(j-1));
- }
- if (plstx.nops() > 0) {
- buffer *= G_numeric(plstx, plsts, 1/p);
- }
- result += buffer;
- }
- return result;
- }
-
// do summation
lst newx;
lst m;
static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
{
- epvector seq;
- seq.push_back(expair(Li(m, x), 0));
- return pseries(rel, seq);
+ if (is_a<lst>(m) || is_a<lst>(x)) {
+ // multiple polylog
+ epvector seq;
+ seq.push_back(expair(Li(m, x), 0));
+ return pseries(rel, seq);
+ }
+
+ // classical polylog
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(s,i) / pow(numeric(i), m);
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
}
prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
// [Kol] (5.3)
- if ((cln::realpart(value) < -0.5) || (n == 0)) {
+ if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95))) {
cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
* cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
{
- epvector seq;
- seq.push_back(expair(S(n, p, x), 0));
- return pseries(rel, seq);
+ if (p == _ex1) {
+ return Li(n+1, x).series(rel, order, options);
+ }
+
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ // subsum = Euler-Zagier-Sum is needed
+ // dirty hack (slow ...) calculation of subsum:
+ std::vector<ex> presubsum, subsum;
+ subsum.push_back(0);
+ for (int i=1; i<order-1; ++i) {
+ subsum.push_back(subsum[i-1] + numeric(1, i));
+ }
+ for (int depth=2; depth<p; ++depth) {
+ presubsum = subsum;
+ for (int i=1; i<order-1; ++i) {
+ subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
+ }
+ }
+
+ for (int i=1; i<order; ++i) {
+ ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
+ }
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
}
}
+// do integration [ReV] (49)
+// put parameter 1 in front of existing parameters
+ex trafo_H_prepend_one(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (int i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(1);
+ return e.subs(h == H(newparameter, h.op(1)).hold());
+ } else {
+ return e * H(lst(1),1-arg).hold();
+ }
+}
+
+
// do integration [ReV] (55)
// put parameter -1 in front of existing parameters
ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
}
+// do x -> 1-x transformation
+struct map_trafo_H_1mx : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(1);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ } else if (parameter.op(0) == -1) {
+ throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
+ } else {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(0);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+ map_trafo_H_1mx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res -= trafo_H_prepend_one(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_prepend_one(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1mx recursion;
+ map_trafo_H_mult unify;
+ ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
+ int firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+ lst newparameter;
+ int j=0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ res = recursion(res).expand() / firstzero;
+ return unify(res);
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
// do x -> 1/x transformation
struct map_trafo_H_1overx : public map_function
{
return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
}
// ... and expand parameter notation
+ bool has_minus_one = false;
lst m;
for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
if (*it > 1) {
m.append(0);
}
m.append(1);
- } else if (*it < -1) {
+ } else if (*it <= -1) {
for (ex count=*it+1; count < 0; count++) {
m.append(0);
}
m.append(-1);
+ has_minus_one = true;
} else {
m.append(*it);
}
}
- // since the transformations produce a lot of terms, they are only efficient for
- // argument near one.
- // no transformation needed -> do summation
+ // do summation
if (cln::abs(x) < 0.95) {
lst m_lst;
lst s_lst;
}
}
+ symbol xtemp("xtemp");
ex res = 1;
// ensure that the realpart of the argument is positive
}
}
- // choose transformations
- symbol xtemp("xtemp");
- if (cln::abs(x-1) < 1.4142) {
- // x -> (1-x)/(1+x)
- map_trafo_H_1mxt1px trafo;
- res *= trafo(H(m, xtemp));
- } else {
- // x -> 1/x
+ // x -> 1/x
+ if (cln::abs(x) >= 2.0) {
map_trafo_H_1overx trafo;
res *= trafo(H(m, xtemp));
if (cln::imagpart(x) <= 0) {
} else {
res = res.subs(H_polesign == I*Pi);
}
+ return res.subs(xtemp == numeric(x)).evalf();
+ }
+
+ // check transformations for 0.95 <= |x| < 2.0
+
+ // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
+ if (cln::abs(x-9.53) <= 9.47) {
+ // x -> (1-x)/(1+x)
+ map_trafo_H_1mxt1px trafo;
+ res *= trafo(H(m, xtemp));
+ } else {
+ // x -> 1-x
+ if (has_minus_one) {
+ map_trafo_H_convert_to_Li filter;
+ return filter(H(m, numeric(x)).hold()).evalf();
+ }
+ map_trafo_H_1mx trafo;
+ res *= trafo(H(m, xtemp));
}
-
- // simplify result
-// TODO
-// map_trafo_H_convert converter;
-// res = converter(res).expand();
-// lst ll;
-// res.find(H(wild(1),wild(2)), ll);
-// res.find(zeta(wild(1)), ll);
-// res.find(zeta(wild(1),wild(2)), ll);
-// res = res.collect(ll);
return res.subs(xtemp == numeric(x)).evalf();
}
if (y.is_zero()) {
return _ex_1_2;
}
- if (y.is_equal(_num1)) {
+ if (y.is_equal(*_num1_p)) {
return zeta(m).hold();
}
if (y.info(info_flags::posint)) {
if (y.info(info_flags::odd)) {
return zeta(m).hold();
} else {
- return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
+ return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
}
} else {
if (y.info(info_flags::odd)) {
- return -bernoulli(_num1-y) / (_num1-y);
+ return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
} else {
return _ex0;
}
git://www.ginac.de / ginac.git / blobdiff